What physical situation will give a negative mass squared?

Question

Consider the Lagrangian density of $\phi^4$ theory $$\mathcal{L}=-\frac{1}{2}\partial^\mu\phi\partial_\mu\phi-\frac{1}{2}m^2\phi^2-\frac{1}{24}\lambda\phi^4$$ the potential energy term is given by $$V(\phi)=\frac{1}{2}m^2\phi^2+\frac{1}{24}\lambda\phi^4$$ when $m^2<0$, the potential will turn to a "W"-like shape. This is a typical situation in spontaneous symmetry breaking.

My question is: under what physical circumstances will we have $m^2<0$? I heard that cooper coupling was a sort of example, is there any good reference on that?

Answer 1

Before electroweak symmetry breaking, the scalar potential in the Standard Electroweak theory is supposed to be described by a potential (similar to $\phi^4$-theory) $$V(\Phi)=m^2(\Phi^\dagger\Phi)+\lambda(\Phi^\dagger\Phi)^2$$ with $m^2<0$ and $\Phi$ being the scalar Higgs doublet. When $m^2<0$, it is just a parameter of the Lagrangian; it does not represent the mass, and therefore, there is nothing wrong about it being negative. After the symmetry breaks spontaneously, the mass of the Higgs field will be given by $=2\lambda v^2=-2m^2$ which is positive. Here, $v$ stands for vacuum expectation value given by $v^2=\frac{-m^2}{\lambda}$. Please verify the numerical factors.

Addendum There is yet another way to understand this. Consider the Lagrangian $$\mathscr{L}=\frac{1}{2}(\partial_\mu\phi)(\partial^\mu\phi)-V(\phi).$$ Let us expand $V(\phi)$ in a Taylor series about a point $\phi=\phi_0$. This gives $$V(\phi)=V(\phi_0)+\frac{\partial V}{\partial\phi}\Big|_{\phi=\phi_0}(\phi-\phi_0)+\frac{1}{2!}\frac{\partial^2 V}{\partial\phi^2}\Big|_{\phi=\phi_0}(\phi-\phi_0)^2+...$$ For the potential $V(\phi)=\frac{1}{2}m^2\phi^2+\lambda\phi^4$, the second derivative of the potential is negative of $m^2<0$. This implies that the potential is expanded not about a minimum but about the maximum $\phi=0$.

Answer 2

I think it is totally possible that there's a more "complete theory" in the UV with a positive mass squared term that, under the RG flow, the mass term turns negative in the IR (e.g. at the electroweak scale).

For example, in this paper, the authors show a similar scenario: the Higgs mass squared term at high energy is originally positive. In the RG flow, the mass term receives contributions from various couplings and becomes negative at lower energy scales.

This cannot happen in the standard model, but can happen in SUSY theories.